Abstract

The periodic nonlinearity that arises from nonideal laser sources and imperfections of optical components limits the accuracy of displacement measurements in heterodyne interferometry at the nanometer level. An analytical approach to investigating the nonlinearity is presented. Frequency mixing, polarization mixing, polarization–frequency mixing, and ghost reflections are all included in this investigation. A general form for the measurement signal, including that of the distortions, is given. The analytical approach is also applicable to homodyne interferometry.

© 1998 Optical Society of America

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References

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  1. R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Packard J. 34, 10 (1983).
  2. C. M. Sutton, “Nonlinearity in length measurements using heterodyne laser Michelson interferometry,” J. Phys. E 20, 1290–1292 (1987).
    [CrossRef]
  3. N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. 26, 2676–2681 (1987).
    [CrossRef] [PubMed]
  4. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
    [CrossRef]
  5. Y. Xie, Y.-Z. Wu, “Elliptical polarization and nonorthogonality of stabilized Zeeman laser output,” Appl. Opt. 28, 2043–2046 (1989).
    [CrossRef] [PubMed]
  6. Y. Xie, Y.-Z. Wu, “Zeeman laser interferometer errors for high-precision measurements,” Appl. Opt. 31, 881–884 (1992).
    [CrossRef] [PubMed]
  7. A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precision Eng. 12, 7–11 (1990).
    [CrossRef]
  8. A. Bergamin, G. Cavagnero, G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. 39, 2053–2074 (1992).
    [CrossRef]
  9. C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
    [CrossRef]
  10. W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precision Eng. 14, 91–98 (1992).
    [CrossRef]
  11. J. A. Stone, L. P. Howard, “A simple technique for observing periodic nonlinearities in Michelson interferometers,” Precision Eng. 22 (1998) in press.
  12. W. Hou, X. Zhao, “Drift of nonlinearity in the heterodyne interferometers,” Precision Eng. 16, 25–35 (1994).
    [CrossRef]
  13. M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt. 35, 1813–1820 (1988).
    [CrossRef]
  14. S. Patterson, J. Beckwith, “Reduction of systematic errors in heterodyne interferometric displacement measurement,” in International Progress in Precision Engineering, Procedings of the 8th International Precision Engineering Seminar, M. Bonis ed. (Elsevier, Amsterdam, 1995) pp. 101–104.

1998 (1)

J. A. Stone, L. P. Howard, “A simple technique for observing periodic nonlinearities in Michelson interferometers,” Precision Eng. 22 (1998) in press.

1996 (1)

C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

1994 (1)

W. Hou, X. Zhao, “Drift of nonlinearity in the heterodyne interferometers,” Precision Eng. 16, 25–35 (1994).
[CrossRef]

1993 (1)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

1992 (3)

Y. Xie, Y.-Z. Wu, “Zeeman laser interferometer errors for high-precision measurements,” Appl. Opt. 31, 881–884 (1992).
[CrossRef] [PubMed]

A. Bergamin, G. Cavagnero, G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. 39, 2053–2074 (1992).
[CrossRef]

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precision Eng. 14, 91–98 (1992).
[CrossRef]

1990 (1)

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precision Eng. 12, 7–11 (1990).
[CrossRef]

1989 (1)

1988 (1)

M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt. 35, 1813–1820 (1988).
[CrossRef]

1987 (2)

C. M. Sutton, “Nonlinearity in length measurements using heterodyne laser Michelson interferometry,” J. Phys. E 20, 1290–1292 (1987).
[CrossRef]

N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. 26, 2676–2681 (1987).
[CrossRef] [PubMed]

1983 (1)

R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Packard J. 34, 10 (1983).

Beckwith, J.

S. Patterson, J. Beckwith, “Reduction of systematic errors in heterodyne interferometric displacement measurement,” in International Progress in Precision Engineering, Procedings of the 8th International Precision Engineering Seminar, M. Bonis ed. (Elsevier, Amsterdam, 1995) pp. 101–104.

Bergamin, A.

A. Bergamin, G. Cavagnero, G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. 39, 2053–2074 (1992).
[CrossRef]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precision Eng. 12, 7–11 (1990).
[CrossRef]

N. Bobroff, “Residual errors in laser interferometry from air turbulence and nonlinearity,” Appl. Opt. 26, 2676–2681 (1987).
[CrossRef] [PubMed]

Cavagnero, G.

A. Bergamin, G. Cavagnero, G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. 39, 2053–2074 (1992).
[CrossRef]

Hou, W.

W. Hou, X. Zhao, “Drift of nonlinearity in the heterodyne interferometers,” Precision Eng. 16, 25–35 (1994).
[CrossRef]

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precision Eng. 14, 91–98 (1992).
[CrossRef]

Howard, L. P.

J. A. Stone, L. P. Howard, “A simple technique for observing periodic nonlinearities in Michelson interferometers,” Precision Eng. 22 (1998) in press.

Mana, G.

A. Bergamin, G. Cavagnero, G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. 39, 2053–2074 (1992).
[CrossRef]

Patterson, S.

S. Patterson, J. Beckwith, “Reduction of systematic errors in heterodyne interferometric displacement measurement,” in International Progress in Precision Engineering, Procedings of the 8th International Precision Engineering Seminar, M. Bonis ed. (Elsevier, Amsterdam, 1995) pp. 101–104.

Player, M. A.

M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt. 35, 1813–1820 (1988).
[CrossRef]

Quenelle, R. C.

R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Packard J. 34, 10 (1983).

Rosenbluth, A. E.

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precision Eng. 12, 7–11 (1990).
[CrossRef]

Stone, J. A.

J. A. Stone, L. P. Howard, “A simple technique for observing periodic nonlinearities in Michelson interferometers,” Precision Eng. 22 (1998) in press.

Su, C. S.

C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

Sutton, C. M.

C. M. Sutton, “Nonlinearity in length measurements using heterodyne laser Michelson interferometry,” J. Phys. E 20, 1290–1292 (1987).
[CrossRef]

Wilkening, G.

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precision Eng. 14, 91–98 (1992).
[CrossRef]

Wu, C. M.

C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

Wu, Y.-Z.

Xie, Y.

Zhao, X.

W. Hou, X. Zhao, “Drift of nonlinearity in the heterodyne interferometers,” Precision Eng. 16, 25–35 (1994).
[CrossRef]

Appl. Opt. (3)

Hewlett Packard J. (1)

R. C. Quenelle, “Nonlinearity in interferometer measurements,” Hewlett Packard J. 34, 10 (1983).

J. Mod. Opt. (2)

A. Bergamin, G. Cavagnero, G. Mana, “Phase holonomy in optical interferometry,” J. Mod. Opt. 39, 2053–2074 (1992).
[CrossRef]

M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt. 35, 1813–1820 (1988).
[CrossRef]

J. Phys. E (1)

C. M. Sutton, “Nonlinearity in length measurements using heterodyne laser Michelson interferometry,” J. Phys. E 20, 1290–1292 (1987).
[CrossRef]

Meas. Sci. Technol. (2)

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

C. M. Wu, C. S. Su, “Nonlinearity in measurements of length by optical interferometry,” Meas. Sci. Technol. 7, 62–68 (1996).
[CrossRef]

Precision Eng. (4)

W. Hou, G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precision Eng. 14, 91–98 (1992).
[CrossRef]

J. A. Stone, L. P. Howard, “A simple technique for observing periodic nonlinearities in Michelson interferometers,” Precision Eng. 22 (1998) in press.

W. Hou, X. Zhao, “Drift of nonlinearity in the heterodyne interferometers,” Precision Eng. 16, 25–35 (1994).
[CrossRef]

A. E. Rosenbluth, N. Bobroff, “Optical sources of nonlinearity in heterodyne interferometers,” Precision Eng. 12, 7–11 (1990).
[CrossRef]

Other (1)

S. Patterson, J. Beckwith, “Reduction of systematic errors in heterodyne interferometric displacement measurement,” in International Progress in Precision Engineering, Procedings of the 8th International Precision Engineering Seminar, M. Bonis ed. (Elsevier, Amsterdam, 1995) pp. 101–104.

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Figures (6)

Fig. 1
Fig. 1

Typical structure of a heterodyne interferometer. BS, beam splitter; other abbreviations and notation defined in text.

Fig. 2
Fig. 2

Three-dimensional figure showing orientations of the laser output and the PBS. The ellipticities of the beams have been exaggerated.

Fig. 3
Fig. 3

Contaminations that are due to both nonorthogonality and misalignment. X and Y represent the system’s reference axes determined by the PBS shown in Fig. 2.

Fig. 4
Fig. 4

Contaminations that are due to ellipticity. The approximation is made that the ellipses of both the measurement and the reference beams are free from eccentricity.

Fig. 5
Fig. 5

Contamination that is due to polarization leakage. Only the measurement beam is illustrated.

Fig. 6
Fig. 6

Summary of error sources in heterodyne interferometry. Frequency mixing, polarization mixing, polarization–frequency mixing, and the ghost reflections are all included.

Equations (12)

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I m     AB   cos Δ ω t + ϕ ,
I r     AB   cos Δ ω t ,
B   cos   θ 2 exp i ω 2 t + α f exp i ω 1 t ,
A   cos   θ 1 exp i ω 1 t + β f exp i ω 2 t ,
α pf   exp i ω 1 t + π / 2
β pf   exp i ω 2 t + π / 2
β p exp i ω 2 t ,
α p exp i ω 1 t ,
E s = A ¯   exp i ω 1 t + β f exp i ω 2 t + α p exp i ω 1 t + β pf   exp i ω 2 t + π / 2 ,
E p = B ¯   exp i ω 2 t + α f exp i ω 1 t + β p exp i ω 2 t + α pf   exp i ω 1 t + π / 2 ,
I m   E s + E p E s + E p *   A ¯ B ¯   cos Δ ω t + ϕ + A ¯ β + B ¯ α cos Δ ω t   + α β + β pf α pf cos Δ ω t - ϕ   + A ¯ β pf - B ¯ α pf sin Δ ω t + α β pf - β α pf   × sin Δ ω t - ϕ ,
α = α p + α f ,     β = β p + β f .

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